Example 1

Determine the prime power decomposition of 273.

Our first prime, 2 does not divide 273. However 3 | 273. Thus we obtain 273 = 3 * 91. 91 is no longer divisible by 3, and is also not divisible by 4, 5, or 6. 7 | 91 though, hence we obtain 273 = 3 * (7 * 13). Note that 3, 7, and 13 are all primes, so we are done.

Where the set of p starts with p1 being the smallest prime number between n and m when they are decomposed, and pk is the largest prime between the integers n and m when they're decomposed. Also e1 ≥ 0, and f1 ≥ 0. Thus, it follows that for the set of primes P = {2, 3, 5, 7, …}, pi and pi+1 are relatively consecutive.

Example 3

Determine the greatest common factor of 273 and 1008.

We found in examples 1 and 2, that the prime decompositions of 273 and 1008 are:

273 = 31 * 71 * 131
1008 = 24 * 32 * 71

Let's rewrite these prime power decompositions in successive primes starting at the minimum prime between 273 and 1008, and ending at the greatest prime between 273 and 1008. These are 2 and 13 respectively. We thus will obtain: